# Time-Reversal Symmetry and Arrow of Time in Quantum Mechanics of Open Systems

## TL;DR

This paper explores how open quantum systems can exhibit broken time-reversal symmetry through resonant states with complex eigenvalues, providing insights into the quantum arrow of time and the emergence of irreversibility.

## Contribution

It demonstrates that eigenstates breaking time-reversal symmetry exist in open quantum systems and explains their role in the emergence of the arrow of time.

## Key findings

- Resonant and anti-resonant states have complex eigenvalues.
- These states observe probability conservation in a specific way.
- The dynamics are dominated by resonant states for t>0, explaining irreversibility.

## Abstract

It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schr\"{o}dinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical," they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for $t>0$; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamics ending at a time-reversal symmetric final state is dominated by the anti-resonant states for $t<0$.